318 



ELECTROSTATICS. 



[PT. II. CH. VII. 



By means of it we may show that the four functions 

 cosh z, sinh z, cos z, sin z 



give conformal representations which are identical except for inter- 

 changes of the axes. By means of the equations 160 A, (i5),(i6), 

 (15'), (16') we obtain the pairs of functions u, v for the four cases, 

 and by the use of equations (3'), (3) after division by one of the 

 factors on the right in the values of u, v we get 



I. w = cosh z, 



u = cosh x cos y, v = sinh x sin y ; 



v 2 



0) 



cos 2 y sin 2 y 



cosh 2 # sinh 2 x 

 II. w = sinh #, 



^ = sinh x cos y, v = cosh x sin y ; 



U? V 2 _ u? 



\) ~:_i~9 ~f~ t.?. = ** 



1. 



III. 



COS , 



u = cos a? cosh 



cos 2 y sin 2 

 v sin x sinh y ; 



= -1. 



() 



1 



x sin' 5 



(a) 



cosh 2 y sinh 5 

 = sin z y 



u = sin x cosh y, v = cos x sinh T/ ; 



/it 2 /ii2 /ii 2 



-1, 



= 1. 



cosh 2 ;?/ sinh 2 y 



sin 2 x cos 2 



(n) -=- cosh a; = sinh x. 

 dx 



(12) j^ 



( 1 3) sinh (ix) = i sin #. 



( 14) cosh (105) = cos x. 



(15) sinh (# + t?/)=:sinhCOS2/ 



+ icosh x sin?/. 



(16) cosh (x + iy) = cosh x cosy 



+ i sinh x sin?/. 



. tanh# + i tan?/ 

 h ^~l + itanha;tan?/ 

 sinh x cosh # + i sin ?/ cos ?/ 

 cos 2 ?/ cosh 2 a; + sin 2 y sinh 2 a; ' 



(17) 



fn') 3- cos # = -sin . 

 k y dx 



(i'2 f ) -r- tan a; = sec 2 x. 

 dx 



(13') sin (ix) = i sinh x. 



( 1 4') cos (ix) = cosh #. 



(15') sin (x + iy) = sin x cosh ?y 



+ icosz sinhy. 

 (16') cos (x + iy) = cosx cosh?/ 



- i sin x sinh y. 

 tan # + i tanh y 



_ sin x cos # + i sinh ?y cosh y 

 ~~ cos 2 a; cosh 2 y + sin 2 a: sinh 2 y ' 



